Singularly Perturbed Monotone Systems and an Application to Double Phosphorylation Cycles

Transcription

1 J Nonlinear Sci (2008) 18: DOI /s Singularly Perturbed Monotone Systems and an Application to Double Phosphorylation Cycles Liming Wang Eduardo D. Sontag Received: 30 July 2007 / Accepted: 28 February 2008 / Published online: 26 March 2008 Springer Science+Business Media LLC 2008 Abstract The theory of monotone dynamical systems has been found very useful in the modeling of some gene protein and signaling networks. In monotone systems every net feedback loop is positive. On the other hand negative feedback loops are important features of many systems since they are required for adaptation and precision. This paper shows that provided that these negative loops act at a comparatively fast time scale the main dynamical property of (strongly) monotone systems convergence to steady states is still valid. An application is worked out to a doublephosphorylation futile cycle motif which plays a central role in eukaryotic cell signaling. Keywords Singular perturbation Monotone systems Asymptotic stability MAPK system Mathematics Subject Classification (2000) 34D15 37C65 1 Introduction Monotone dynamical systems constitute a rich class of models for which global and almost-global convergence properties can be established. They are particu- L. Wang ( ) E.D. Sontag Department of Mathematics Rutgers University 110 Frelinghuysen Road Piscataway NJ USA wshwlm@math.rutgers.edu E.D. Sontag sontag@math.rutgers.edu

2 528 J Nonlinear Sci (2008) 18: Fig. 1 Dual futile cycle. A substrate S 0 is ultimately converted into a product S 2 in an activation reaction triggered or facilitated by an enzyme E and conversely S 2 is transformed back (or deactivated ) into the original S 0 helped on by the action of a second enzyme F larly useful in biochemical applications and also appear in areas like coordination (Moreau 2004) and other problems in control (Chisci and Falugi 2005). One of the fundamental results in monotone systems theory is Hirsch s Generic Convergence Theorem (Hirsch ; Hirsch and Smith 2005; Smith 1995). Informally stated Hirsch s result says that almost every bounded solution of a strongly monotone system converges to the set of equilibria. There is a rich literature regarding the application of this powerful theorem as well as of other results dealing with everywhere convergence when equilibria are unique (Dancer 1998; Jiang 1994; Smith 1995) to models of biochemical systems. See for instance Sontag ( ) for expositions and many references. Unfortunately many models in biology are not monotone at least with respect to any standard orthant order. This is because in monotone systems (with respect to orthant orders) every net feedback loop should be positive; on the other hand in many systems negative feedback loops often appear as well as they are required for adaptation and precision. Intuitively however negative loops that act at a comparatively fast time scale should not affect the main characteristics of monotone behavior. The main purpose of this paper is to show that this is indeed the case in the sense that singularly perturbed strongly monotone systems inherit generic convergence properties. A system that is not monotone may become monotone once fast variables are replaced by their steady-state values. In order to prove a precise time-separation result we employ tools from geometric singular perturbation theory. This point of view is of special interest in the context of biochemical systems; for example Michaelis Menten kinetics are mathematically justified as singularly perturbed versions of mass action kinetics (Edelstein-Keshet 1988; Murray 2002). One particular example of great interest in view of current systems biology research is that of dual futile cycle motifs as illustrated in Fig. 1. As discussed in Samoilov et al. (2005) futile cycles (with any number of intermediate steps and also called substrate cycles enzymatic cycles or enzymatic interconversions) underlie signaling processes such as guanosine triphosphatase cycles (Donovan et al. 2002) bacterial two-component systems and phosphorelays (Bijlsma and Groisman 2003; Grossman 1995) actin treadmilling (Chen et al. 2000) and glucose mobilization (Karp 2002) as well as metabolic control (Stryer 1995) and cell division and apoptosis (Sulis and Parsons 2003) and cell-cycle checkpoint control (Lew and Burke 2003). A most important instance is that of Mitogen-Activated Protein Kinase (MAPK) cascades which regulate primary cellular activities such as proliferation differentiation and apoptosis (Asthagiri and Lauffenburger 2001; Chang and Karin 2001; Huang and Ferrell 1996; Widmann et al. 1999) in eukaryotes from yeast to humans.

3 J Nonlinear Sci (2008) 18: MAPK cascades usually consist of three tiers of similar structures with multiple feedbacks (Burack and Sturgill 1997; Ferrell and Bhatt 1900; Zhao and Zhang 2001). Here we focus on one individual level of a MAPK cascade which is a dual futile cycle as depicted in Fig. 1. The precise mathematical model is described later. Numerical simulations of this model have suggested that the system may be monostable or bistable see Markevich et al. (2004). The latter will give rise to switch-like behavior which is ubiquitous in cellular pathways (Gardner et al. 2000; Pomerening et al. 2003; Sel kov 1975; Sha et al. 2003). In either case the system under meaningful biological parameters shows convergence not other dynamical properties such as periodic behavior or even chaotic behavior. Analytical studies done for the quasi-steady-state version of the model (slow dynamics) which is a monotone system indicate that the reduced system is indeed monostable or bistable see Ortega et al. (2006). Thus it is of great interest to show that at least in certain parameter ranges (as required by singular perturbation theory) the full system inherits convergence properties from the reduced system and this is what we do as an application of our results. We remark that the simplified system consisting of a unary conversion cycle (no S 2 ) is known to admit a unique equilibrium (subject to mass conservation constraints) which is a global attractor see Angeli and Sontag (2008). A feature of our approach is the use of geometric invariant manifold theory (Fenichel 1979; Jones 1995;Nipp1992). There is a manifold M ε invariant for the full dynamics of a singularly perturbed system which attracts all near-enough solutions. However we need to exploit the full power of the theory and especially the fibration structure and an asymptotic phase property. The system restricted to the invariant manifold M ε is a regular perturbation of the slow (ε = 0) system. As remarked in Theorem 1.2 in Hirsch s early paper (Hirsch 1985) a C 1 regular perturbation of a flow with eventually positive derivatives also has generic convergence properties. So solutions in the manifold will generally be well-behaved and asymptotic phase implies that solutions near M ε track solutions in M ε and hence also converge to equilibria if solutions on M ε do. A key technical detail is to establish that the tracking solutions also start from the good set of initial conditions for generic solutions of the large system. A preliminary version of these results in Wang and Sontag (2006) dealt with the special case of singularly perturbed systems of the form: ẋ =f(xy) εẏ =Ay + h(x) on a product domain where A is a constant Hurwitz matrix and the reduced system ẋ = f(x A 1 h(x)) is strongly monotone. However for the application to the above futile cycle there are two major problems with that formulation: first the dynamics of the fast system have to be allowed to be nonlinear in y and second it is crucial to allow for an ε-dependence on the right-hand side as well as to allow the domain to be a convex polytope depending on ε. We provide a much more general formulation here. We note that no assumptions are imposed regarding global convergence of the reduced system which is essential because of the intended application to multistable

4 530 J Nonlinear Sci (2008) 18: systems. This seems to rule out the applicability of Lyapunov-theoretic and input-tostate stability tools (Christofides and Teel 1996; Teel et al. 2003). This paper is organized as follows. The main result is stated in Sect. 2. In Sect. 3 we review some basic definitions and theorems about monotone systems. The detailed proof of the main theorem can be found in Sect. 4 and applications to the MAPK system and another set of ordinary differential equations are discussed in Sect. 5. Finally in Sect. 6 we summarize the key points of this paper. 2 Statement of the Main Theorem In this paper we focus on the dynamics of the following prototypical system in singularly perturbed form: dx dt = f 0(xyε) ε dy dt = g 0(xyε). We will be interested in the dynamics of this system on an ε-dependent domain D ε. For 0 <ε 1 the variable x changes much slower than y. As long as ε 0 one may also change the time scale to τ = t/ε and study the equivalent form: (1) dx dτ = εf 0(xyε) dy dτ = g 0(xyε). (2) Within this general framework we will make the following assumptions (some technical terms will be defined later) where the integer r>1 and the positive number ε 0 are fixed from now on: A1 Let U R n and V R m be open and bounded. The functions and f 0 : U V [0ε 0 ] R n g 0 : U V [0ε 0 ] R m are both of class C r b where a function f is in Cr b if it is in Cr and its derivatives up to order r as well as f itself are bounded. A2 There is a function m 0 : U V of class C r b such that g 0(x m 0 (x) 0) = 0 for all x in U.

5 J Nonlinear Sci (2008) 18: It is often helpful to consider z = y m 0 (x) and the fast system (2) in the new coordinates becomes: dx dτ = εf 1(xzε) (3) dz dτ = g 1(xzε) where f 1 (xzε)= f 0 ( xz + m0 (x) ε ) g 1 (xzε)= g 0 ( xz + m0 (x) ε ) ε [ D x m 0 (x) ] f 1 (xzε). When ε = 0 the system (3) degenerates to dz dτ = g 1(x z 0) x(τ) x 0 U (4) seen as equations on {z z + m 0 (x 0 ) V }. A3 The steady state z = 0of(4) is globally asymptotically stable on {z z+m 0 (x 0 ) V } for all x 0 U. A4 All eigenvalues of the matrix D y g 0 (x m 0 (x) 0) have negative real parts for every x U i.e. the matrix D y g 0 (x m 0 (x) 0) is Hurwitz on U. (The notation D y g 0 (x m 0 (x) 0) means the partial derivatives of g 0 (xyε) with respect to y evaluated at the point (x m 0 (x) 0).) A5 There exists a family of convex compact sets D ε U V which depend continuously on ε [0ε 0 ] such that (1) is positively invariant on D ε for ε (0ε 0 ]. A6 The flow ψ 0 t of the limiting system (set ε = 0in(1)): dx dt = f ( 0 xm0 (x) 0 ) (5) has eventually positive derivatives on K 0 with respect to some cone where K 0 is the projection of D 0 { (x y) y = m 0 (x) x U } onto the x-axis. A7 The set of equilibria of (1) ond ε is totally disconnected. Remark 1 Assumption A3 implies that y = m 0 (x) is a unique solution of g 0 (x y 0) = 0onU. Continuity in A5 is understood with respect to the Hausdorff metric. In mass-action chemical kinetics the vector fields are polynomials. So A1 follows naturally. Our main theorem is:

6 532 J Nonlinear Sci (2008) 18: Theorem 1 Under assumptions A1 to A7 there exists a positive constant ε <ε 0 such that for each ε (0ε ) the forward trajectory of (1) starting from almost every point in D ε converges to some equilibrium. 3 Monotone Systems of Ordinary Differential Equations In this section we review several useful definitions and theorems regarding monotone systems. As we wish to provide results valid for arbitrary orders not merely orthants and some of these results though well-known are not readily available in a form needed for reference we provide some technical proofs. Definition 1 A nonempty closed set C R N is a cone if 1. C + C C 2. R + C C 3. C ( C) ={0}. We always assume C {0}. Associated to a cone C is a partial order on R N.For any xy R N we define When Int C is not empty we can define x y x y C x>y x y C x y. x y x y Int C. Definition 2 The dual cone of C is defined as An immediate consequence is C = { λ ( R N ) λ(c) 0 }. x C λ(x) 0 λ C x Int C λ(x) > 0 λ C \{0}. With this partial ordering on R N we analyze certain features of the dynamics of an ordinary differential equation: dz dt = F(z) (6) where F : R N R N is a C 1 vector field. We are interested in a special class of equations that preserve the ordering along the trajectories. For simplicity the solutions of (6) are assumed to exist for all t 0 in the sets considered in the following.

7 J Nonlinear Sci (2008) 18: Definition 3 The flow φ t of (6) is said to have (eventually) positive derivatives on asetv R N if[d z φ t (z)]x Int C for all x C \{0}z V and t 0(t t 0 for some t 0 > 0 independent of z). It is worth noticing that [D z φ t (z)]x Int C is equivalent to λ([d z φ t (z)]x) > 0for all λ C with norm one. We will use this fact in the proof of the next lemma which deals with regular perturbations in the dynamics. The proof is in the same spirit as in Theorem 1.2 of Hirsch (1983) but generalized to the arbitrary cone C. Lemma 1 Assume V R N is a compact set in which the flow φ t of (6) has eventually positive derivatives. Then there exists δ>0 with the following property. Let ψ t denote the flow of a C 1 vector field G such that the C 1 norm of F(z) G(z) is less than δ for all z in V. Then there exists t > 0 such that if ψ s (z) V for all s [0t] where t t then [D z ψ t (z)]x Int C for all x C \{0}. Proof Pick t = t 0 > 0 so that λ([d z φ t (z)]x) > 0 for all t t 0 z Vλ C x C with λ =1 x =1. Then there exists δ>0 with the property that when the C 1 norm of F(z) G(z) is less than δwehaveλ([d z ψ t (z)]x) > 0fort 0 t 2t 0. When t>2t 0 we write t = r + kt 0 where t 0 r<2t 0 and k N. Ifψ s (z) V for all s [0t] we can define z j := ψ jt0 (z) for j = 0...k. For any x C \{0} using the chain rule we have: [ Dz ψ t (z) ] x = [ D z ψ r (z k ) ][ D z ψ t0 (z k 1 ) ] [D z ψ t0 (z 0 ) ] x. It is easy to see that [D z ψ t (z)]x Int C. Corollary 1 If V is positively invariant under the flow ψ t then ψ t has eventually positive derivatives in V. Proof If V is positively invariant under the flow ψ t then for any z V the condition ψ s (z) V for s [0t] is satisfied for all t 0. By the previous lemma ψ t has eventually positive derivatives in V. Definition 4 The system (6) ortheflowφ t of (6) is called monotone (resp. strongly monotone) in a set W R N if for all t>0 and z 1 z 2 W z 1 z 2 φ t (z 1 ) φ t (z 2 ) ( resp. φt (z 1 ) φ t (z 2 ) when z 1 z 2 ). It is eventually (strongly) monotone if there exists t 0 > 0 such that φ t is (strongly) monotone for all t t 0. Definition 5 An set W R N is called p-convex if W contains the entire line segment joining x and y whenever x y xy W. Proposition 1 Let W R N be p-convex. If the flow φ t has (eventually) positive derivatives in W then it is (eventually) strongly monotone in W.

8 534 J Nonlinear Sci (2008) 18: Proof For any z 1 >z 2 Wλ C \{0} and t 0(t t 0 for some t 0 > 0) we have that λ(φ t (z 1 ) φ t (z 2 )) equals 1 0 λ ([ D z φ t (sz 1 + (1 s)z 2 ) ] (z 1 z 2 ) ) ds > 0. Therefore φ t is (eventually) strongly monotone in W. The following two lemmas are variations of Hirsch s Generic Convergence Theorem. Lemma 2 Suppose that the flow φ t of (6) has eventually positive derivatives in a p- convex open set W R N. Let W c W be the set of points whose forward orbit has compact closure in W. If the set of equilibria is totally disconnected (e.g. countable) then the forward trajectory starting from almost every point in W c converges to an equilibrium. This result follows from a generalization of Theorem 4.1 in Hirsch (1985) toan arbitrary cone. Definition 6 A point x inasetw R N is called strongly accessible from below (resp. above) if there exists a sequence {y n } in W converging to x such that y n < y n+1 <x(resp. y n >y n+1 >x). In our motivating example as well as in most biochemical systems after reduction by elimination of stoichiometric constraints the set of equilibria is discrete and thus Lemma 2 will apply. However the following more general result is also true and applies even when the set of equilibria is not discrete. This follows as a direct application of Theorem 2.26 in Hirsch and Smith (2005). Lemma 3 Suppose that the flow φ t of (6) has compact closure and eventually positive derivatives in a p-convex open set W R N. If any point in W can be strongly accessible either from above or from below in W then the forward trajectory from every point except for initial conditions in a nowhere dense set converges to an equilibrium. 4 Details of the Proof Our approach to solve the varying domain problem is motivated by Nipp (1992). The idea is to extend the vector fields from U V [0ε] to R n R m [0ε 0 ] then apply geometric singular perturbation theorems (Sakamoto 1990)onR n R m [0ε 0 ] and finally restrict the flows to D ε for the generic convergence result.

9 J Nonlinear Sci (2008) 18: Extensions of the Vector Fields For a given compact set K R n (K 0 K U) the following procedure is adopted from Nipp (1992) to extend a Cb r function with respect to the x coordinate from U to R n such that the extended function is Cb r and agrees with the old one on K.Thisisa routine smooth patching argument. Let U 1 be an open subset of U with C r boundary and such that K U 1 U.For 0 > 0 sufficiently small define U 0 1 := { x U 1 (x) 0 } where (x) := min u U 1 x u such that K is contained in U 0 1. Consider the scalar C function ρ: 0 a 0 ρ(a) := exp(1 exp(a 1)/a) 0 <a<1 1 a 1. Define and For any q Cb r (U) let 0 x R n \ U 1 ˆ (x) := (x) x U 1 \ U 0 0 x U 0 1 ( ) ˆ (x) (x) := ρ. 0 1 { q(x) x U1 q(x) := 0 x R n \ U 1 and q(x) := (x) q(x). Then q(x) Cb r(rn ) and q(x) q(x) on K. We fix some d 0 > 0 such that L d0 := { z R m } { z d 0 z z + m0 (x) V }. Then we extend the functions f 1 and m 0 to f 1 and m 0 respectively with respect to x in the above way. To extend g 1 let us first rewrite the differential equation for z as where x K dz dτ = [ B(x) + C(xz) ] z + εh(xzε) ε [ D x m 0 (x) ] f 1 (xzε) B(x) = D y g 0 ( xm0 (x) 0 ) and C(x0) = 0.

10 536 J Nonlinear Sci (2008) 18: Following the above procedures we extend the functions C and H to C and H but the extension of B is defined as B(x) := (x) B(x) μ ( 1 (x) ) I n where μ is the positive constant such that the real parts of all eigenvalues of B(x) is less than μ for every x K. According to the definition of B(x) all eigenvalues of B(x) will have negative real parts less than μ for every x R n. The extension ḡ 1 defined as [ B(x)+ C(xz) ] z + εh(xzε) ε [ D x m 0 (x) ] f 1 (xzε) is then Cb r 1 (R n L d0 [0ε 0 ]) and agrees with g 1 on K L d0 [0ε 0 ]. To extend functions f 1 and ḡ 1 in the z direction from L d0 to R m weusethesame extension technique but with respect to z. Let us denote the extensions of f 1 C H and the function z = z by f 1 C H and z respectively then define g 1 as [ B(x) + C(xz) ] z(z) + εh(xzε) ε [ D x m 0 (x) ] f 1 (xzε) which is now Cb r 1 (R n R m [0ε 0 ]) and agrees with g 1 on K L d1 [0ε 0 ] for some d 1 slightly less than d 0. Notice that z = 0 is a solution of g 1 (x z 0) = 0 which guarantees that for the extended system in (x y) coordinates (y = z + m 0 (x)) dx dτ = εf(xyε) (7) dy dτ = g(xyε) y = m 0 (x) is the solution of g(xy0) = 0. To summarize (7) satisfies E1 The functions f Cb( r R n R m [0ε 0 ] ) g r 1 ( R n R m [0ε 0 ] ) m 0 Cb( r R n ) g ( x m 0 (x) 0 ) = 0 x R n. E2 All eigenvalues of the matrix D y g(x m 0 (x) 0) have negative real parts less than μ for every x R n. E3 The function m 0 coincides with m 0 on K and the functions f and g coincide with f 0 and g 0 respectively on d1 := { (x y) x K y m 0 (x) d 1 }. Conditions E1 and E2 are the assumptions for geometric singular perturbation theorems and condition E3 ensures that on d1 the flow of (2) coincides with the flow of (7). If we apply geometric singular perturbation theorems to (7) on R n R m [0ε 0 ] the exact same results are true for (2) on d1. For the rest of the paper we identify the flow of (7) and the flow of (2)on d1 without further mentioning this fact. (Later in Lemmas 5 8 when globalizing the results we consider again the original system.)

11 J Nonlinear Sci (2008) 18: Geometric Singular Perturbation Theory The theory of geometric singular perturbation can be traced back to the work of Fenichel (1979) which first revealed the geometric aspects of singular perturbation problems. Later on the works by Knobloch and Aulbach (1984) Nipp (1992) and Sakamoto (1990) also presented results similar to Fenichel (1979). By now the theory is fairly standard and there have been enormous applications to traveling waves of partial differential equations see Jones (1995) and the references there. To apply geometric singular perturbation theorems to the vector fields on R n R m [0ε 0 ] we use the theorems stated in Sakamoto (1990). The following lemma is a restatement of the theorems in Sakamoto (1990) and we refer to Sakamoto (1990) for the proof. Lemma 4 Under conditions E1 and E2 there exists a positive ε 1 <ε 0 such that for every ε (0ε 1 ]: 1. There is a Cb r 1 function m : R n [0ε 1 ] R m such that the set M ε defined by M ε := { (x m(x ε)) x R n} is invariant under the flow generated by (7). Moreover sup m(x ε) m0 (x) = O(ε) as ε 0. x R n In particular we have m(x 0) = m 0 (x) for all x R n. 2. The set consisting of all the points (x 0 y 0 ) such that sup y(τ; x 0 y 0 ) m(x(τ; x 0 y 0 ) ε) e μτ 4 < τ 0 where (x(τ; x 0 y 0 ) y(τ; x 0 y 0 )) is the solution of (7) passing through (x 0 y 0 ) at τ = 0 is a C r 1 -immersed submanifold in R n R m of dimension n + m denoted by W s (M ε ) the stable manifold of M ε. 3. There is a positive constant δ 0 such that if sup y(τ; x 0 y 0 ) m(x(τ; x 0 y 0 ) ε) <δ 0 τ 0 then (x 0 y 0 ) W s (M ε ). 4. The manifold W s (M ε ) is a disjoint union of C r 1 -immersed manifolds Wε s (ξ) of dimension m: W s (M ε ) = Wε s (ξ). ξ R n

12 538 J Nonlinear Sci (2008) 18: Fig. 2 An illustration of the positive invariant and asymptotic phase properties. Let p 0 be a point on the fiber W s ε (q 0) (vertical curve). Suppose the solution of (7) starting from q 0 M ε evolves to q 1 M ε at time τ 1 then the solution of (7) starting from p 0 will evolve to p 1 W s ε (q 1) at time τ 1. At time τ 2 theyevolve to q 2 p 2 respectively. These two solutions are always on the same fiber. If we know that the one starting from q 0 converges to an equilibrium then the one starting from p 0 also converges to an equilibrium For each ξ R n let H ε (ξ)(τ) be the solution for τ 0 of dx dτ = εf (x m(x ε) ε) x(0) = ξ Rn. Then the manifold Wε s (ξ) is the set where { (x 0 y 0 ) sup τ 0 x(τ) e μτ 4 < sup ỹ(τ) e μτ 4 < } τ 0 x(τ) = x(τ; x 0 y 0 ) H ε (ξ)(τ) ỹ(τ) = y(τ; x 0 y 0 ) m ( H ε (ξ)(τ) ε ). 5. The fibers are positively invariant in the sense that W s ε (H ε(ξ)(τ)) is the set {( x(τ; x0 y 0 ) y(τ; x 0 y 0 ) ) (x 0 y 0 ) W s ε (ξ)} for each τ 0 see Fig The fibers restricted to the δ 0 neighborhood of M ε denoted by Wεδ s 0 can be parameterized as follows. There are two Cb r 1 functions P εδ0 : R n L δ0 R n Q εδ0 : R m L δ0 R m and a map mapping (ξ η) to (x y) where T εδ0 : R n L δ0 R n R m x = ξ + P εδ0 (ξ η) y = m(x ε) + Q εδ0 (ξ η)

13 J Nonlinear Sci (2008) 18: such that W s εδ 0 (ξ) = T εδ0 (ξ L δ0 ). Remark 2 The δ 0 in property 3 can be chosen uniformly for ε (0ε 0 ]. Without loss of generality we assume that δ 0 <d 1. Notice that property 4 ensures that for each (x 0 y 0 ) W s (M ε ) there exists a ξ such that x(τ; x0 y 0 ) H ε (ξ)(τ) 0 y ( τ; x0 y 0 ) m ( Hε (ξ)(τ) ε ) 0. as τ. This is often referred to as the asymptotic phase property see Fig Further Analysis of the Dynamics The first property of Lemma 4 concludes the existence of an invariant manifold M ε. There are two reasons to introduce M ε.firstonm ε the x-equation is decoupled from the y-equation: dx dt = f ( xm(xε)ε ) y(t) = m ( x(t)ε ) (8). This reduction allows us to analyze a lower dimensional system whose dynamics may have been well studied. Second when ε approaches zero the limit of (8) is(5) on K 0.If(5) has some desirable property it is natural to expect that this property is inherited by (8). An example of this principle is provided by the following lemma: Lemma 5 There exists a positive constant ε 2 <ε 1 such that for each ε (0ε 2 ) the flow ψt ε of (8) has eventually positive derivatives on K ε which is the projection of M ε D ε to the x-axis. Proof Assumption A6 states that the flow ψt 0 of the limiting system (5) has eventually positive derivatives on K 0. By the continuity of m(x ε) and D ε at ε = 0 we can pick ε 2 small enough such that the flow ψt 0 has eventually positive derivatives on K ε for all ε (0ε 2 ). Applying Corollary 1 we conclude that the flow ψt ε of (8) has eventually positive derivatives on K ε provided K ε is positively invariant under (8) which follows easily from the facts that (7) is positively invariant on D ε and M ε is an invariant manifold. The next lemma asserts that the generic convergence property is preserved for (8) see Fig. 3. Lemma 6 For each ε (0ε 2 ) there exists a set C ε K ε such that the forward trajectory of (8) starting from any point of C ε converges to some equilibrium and the Lebesgue measure of K ε \ C ε is zero.

14 540 J Nonlinear Sci (2008) 18: Fig. 3 This is a sketch of the manifolds M 0 (surface bounded by dashed curves) M ε (surface bounded by dotted curves) and D ε (the cube). It highlights two major characteristics of M ε. First M ε is close to M 0. Second the trajectories on M ε converge to equilibria if those on M 0 do Proof There exists a convex open set W ε containing K ε such that flow ψt ε of (8) has eventually positive derivatives on W ε. Assumption A5 assures that K ε Wε c.the proof is completed by applying Lemma 2 under the assumption A7. By now we have discussed flows restricted to the invariant manifold M ε.nextwe will explore the conditions for a point to be on W s (M ε ) the stable manifold of M ε. Property 3 of Lemma 4 provides a sufficient condition namely any point (x 0 y 0 ) such that sup y(τ; x0 y 0 ) m ( x(τ; x 0 y 0 ) ε ) <δ0 (9) τ 0 is on W s (M ε ). In fact if we know that the difference between y 0 and m(x 0 ε) is sufficiently small then the above condition is always satisfied. More precisely we have: Lemma 7 There exists ε 3 > 0δ 0 >d>0 such that for each ε (0ε 3 ) if the initial condition satisfies y 0 m(x 0 ε) <d then (9) holds i.e. (x 0 y 0 ) W s (M ε ). Proof Follows from the proof of Claim 1 in Nipp (1992). Before we get further into the technical details let us give an outline of the proof of the main theorem. The proof can be decomposed into three steps. First we show that almost every trajectory on D ε M ε converges to some equilibrium. This is precisely Lemma 6. Second we show that almost every trajectory starting from W s (M ε ) converges to some equilibrium. This follows from Lemma 6 and the asymptotic phase property in Lemma 4 but we still need to show that the set of nonconvergent initial conditions is of measure zero. The last step is to show that all trajectories in D ε will eventually stay in W s (M ε ) which is our next lemma:

15 J Nonlinear Sci (2008) 18: Lemma 8 There exist positive τ 0 and ε 4 <ε 3 such that (x(τ 0 ) y(τ 0 )) W s (M ε ) for all ε (0ε 4 ) where (x(τ) y(τ)) is the solution to (2) with the initial condition (x 0 y 0 ) D ε. Proof It is convenient to consider the problem in (x z) coordinates. Let (x(τ) z(τ)) bethesolutionto(3) with initial condition (x 0 z 0 ) where z 0 = y 0 m(x 0 0). We first show that there exists a τ 0 such that z(τ 0 ) d/2. Expanding g 1 (xzε)at the point (x 0 z0) the equation of z becomes dz dτ = g 1(x 0 z0) + g 1 x (ξ z 0)(x x 0) + εr(xzε) for some ξ(τ) between x 0 and x(τ) (where ξ(τ) can be picked continuously in τ ). Let us write z(τ) = z 0 (τ) + w(τ) where z 0 (τ) is the solution to (4) with initial the condition z 0 (0) = z 0 and w(τ) satisfies dw dτ = g 1(x 0 z0) g 1 (x 0 z 0 0) + g 1 x (ξ z 0)(x x 0) + εr(xzε) = g 1 z (x 0ζ0)w + ε g τ 1 (ξ z 0) f 1 (x(s) z(s) ε) ds + εr(xzε) (10) x 0 with the initial condition w(0) = 0 and some ζ(τ) between z 0 (τ) and z(τ) (where ζ(τ) can be picked continuously in τ ). By assumption A3 there exist a positive τ 0 such that z 0 (τ) d/4 for all τ τ 0. Notice that we are working on the compact set D ε soτ 0 can be chosen uniformly for all initial conditions in D ε. We write the solution of (10) as τ g 1 w(τ) = 0 z (x 0ζ0)w ds τ ( g1 + ε (ξ z 0) x 0 s 0 ) f 1 (xzε)ds + R(xzε) ds. Since the functions f 1 R and the derivatives of g 1 are bounded on D ε wehave w(τ) τ τ s ) L w ds + ε (M 1 M 2 ds + M 3 ds for some positive constants LM i i = The notation w means the Euclidean norm of w R m. Moreover if we define τ s ) α(τ) = (M 1 M 2 ds + M 3 ds 0 0

16 542 J Nonlinear Sci (2008) 18: then w(τ) τ L w ds + εα(τ 0 ) 0 for all τ [0τ 0 ] as α is increasing in τ. Applying Gronwall s inequality (Sontag 1990) we have w(τ) εα(τ0 )e Lτ which holds in particular at τ = τ 0. Finally we choose ε 4 small enough such that εα(τ 0 )e Lτ 0 <d/4 and m(x ε) m(x 0) <d/2 for all ε (0ε 4 ). Then we have y(τ0 ) m(x(τ 0 ) ε) y(τ0 ) m(x(τ 0 ) 0) + m(x(τ0 ) ε) m(x(τ 0 ) 0) < z(τ 0 ) + d/2 <d/2 + d/2 = d. That is (x(τ 0 ) y(τ 0 )) W s (M ε ) by Lemma 7. By now we have completed all three steps and are ready to prove Theorem Proof of Theorem 1 Proof Let ε = min{ε 2 ε 4 }.Forε (0ε ) it is equivalent to prove the result for the fast system (2). Pick an arbitrary point (x 0 y 0 ) in D ε and there are three cases: 1. y 0 = m(x 0 ε) that is (x 0 y 0 ) M ε D ε. By Lemma 6 the forward trajectory converges to an equilibrium except for a set of measure zero < y 0 m(x 0 ε) <d. By Lemma 7 we know that (x 0 y 0 ) is in W s (M ε ). Then property 4 of Lemma 4 guarantees that the point (x 0 y 0 ) is on some fiber Wεd s (ξ) where ξ K ε.ifξ C ε that is the forward trajectory of ξ converges to some equilibrium then by the asymptotic phase property of Lemma 4 the forward trajectory of (x 0 y 0 ) also converges to an equilibrium. To deal with the case when ξ is not in C ε it is enough to show that the set B εd = Wεd s (ξ) ξ K ε \C ε has measure zero in R m+n. Define S εd = (K ε \ C ε ) L d. By Lemma 6 K ε \ C ε has measure zero in R n thus S εd has measure zero in R n R m. On the other hand property 6 in Lemma 4 implies B εd = T εd (S εd ). Since Lipschitz maps send measure zero sets to measure zero sets B εd is of measure zero. 3. y 0 m(x 0 ε) d. By Lemma 8 the point (x(τ 0 ) y(τ 0 )) is in W s (M ε ) and we are back to case 2. The proof is completed if the set φ τ ε 0 (B εd ) has measure zero where φτ ε is the flow of (2). This is true because φε τ is a diffeomorphism for any finite τ.

17 J Nonlinear Sci (2008) 18: Applications 5.1 An Application to the Dual Futile Cycle Our structure of futile cycles in Fig. 1 implicitly assumes a sequential instead of a random mechanism. By a sequential mechanism we mean that the kinase phosphorylates the substrates in a specific order and the phosphatase works in the reverse order. A few kinases are known to be sequential for example the auto-phosphorylation of FGF-receptor-1 kinase (Furdui et al. 2006). This assumption dramatically reduces the number of different phospho-forms and simplifies our analysis. In a special case when the kinetic constants of each phosphorylation are the same and the kinetic constants of each dephosphorylation are the same the random mechanism can be easily included in the sequential case. We therefore write down the chemical reactions in Fig. 1 as follows: S 0 + E k 1 k 2 C 1 S 1 + E k 3 k 1 S 2 + F h 1 h 2 C 3 S 1 + F h 3 h 1 There are three conservation relations: k 3 C 2 k 4 h 3 C 4 h 4 S 2 + E S 0 + F. S tot =[S 0 ]+[S 1 ]+[S 2 ]+[C 1 ]+[C 2 ]+[C 4 ]+[C 3 ] E tot =[E]+[C 1 ]+[C 2 ] F tot =[F ]+[C 4 ]+[C 3 ] where brackets indicate concentrations. Based on mass action kinetics we have the following set of ordinary differential equations: d[s 0 ] dτ d[s 2 ] dτ d[c 1 ] dτ d[c 2 ] dτ d[c 4 ] dτ d[c 3 ] dτ = h 4 [C 4 ] k 1 [S 0 ][E]+k 1 [C 1 ] = k 4 [C 2 ] h 1 [S 2 ][F ]+h 1 [C 3 ] = k 1 [S 0 ][E] (k 1 + k 2 )[C 1 ] = k 3 [S 1 ][E] (k 3 + k 4 )[C 2 ] = h 3 [S 1 ][F ] (h 3 + h 4 )[C 4 ] = h 1 [S 2 ][F ] (h 1 + h 2 )[C 3 ]. (11)

18 544 J Nonlinear Sci (2008) 18: After rescaling the concentrations and time (11) becomes dx 1 dt dx 2 dt ε dy 1 dt ε dy 2 dt ε dy 3 dt = k 1 S tot x 1 (1 y 1 y 2 ) + k 1 y 1 + h 4 cy 3 = h 1 S tot cx 2 (1 y 3 y 4 ) + h 1 cy 4 + k 4 y 2 = k 1 S tot x 1 (1 y 1 y 2 ) (k 1 + k 2 )y 1 = k 3 S tot (1 x 1 x 2 εy 1 εy 2 εcy 3 εcy 4 ) (1 y 1 y 2 ) (k 3 + k 4 )y 2 = h 3 S tot (1 x 1 x 2 εy 1 εy 2 εcy 3 εcy 4 ) (1 y 3 y 4 ) (h 3 + h 4 )y 3 (12) where ε dy 4 dt = h 1 S tot x 2 (1 y 3 y 4 ) (h 1 + h 2 )y 4 x 1 = [S 0] S tot x 2 = [S 2] S tot y 1 = [C 1] E tot y 2 = [C 2] E tot y 3 = [C 4] F tot y 4 = [C 3] F tot ε= E tot S tot c= F tot E tot t = τε. These equations are in the form of (1). The conservation laws suggest taking ε 0 = 1/(1 + c) and D ε = { (x 1 x 2 y 1 y 2 y 3 y 4 ) 0 y 1 + y y 3 + y 4 1x 1 x 2 y 1 y 2 y 3 y x 1 + x 2 + ε(y 1 + y 2 + cy 3 + cy 4 ) 1 }. For ε (0ε 0 ] taking the inner product of the normal of D ε and the vector fields it is easy to check that (12) is positively invariant on D ε so A5 holds. We want to emphasize that in this example the domain D ε is a convex polytope varying with ε. It can be proved that on D ɛ system (12) has at most a finite number of steady states and thus A7 holds. This is a consequence of a more general result proved using some of the ideas given in Gunawardena (2005) concerning the number of steady states of more general systems of phosphorylation/dephosphorylation reactions see Wang and Sontag (2008). Solving g 0 (x y 0) = 0 we get y 1 = y 2 = x 1 K m1 S tot + K m1(1 x 1 x 2 ) K m2 + x 1 K m1 (1 x 1 x 2 ) K m2 K m1 S tot + K m1(1 x 1 x 2 ) K m2 + x 1

19 J Nonlinear Sci (2008) 18: y 3 = y 4 = K m3 (1 x 1 x 2 ) K m4 K m3 S tot + K m3(1 x 1 x 2 ) K m4 + x 2 x 2 K m3 S tot + K m3(1 x 1 x 2 ) K m4 + x 2 where K m1 K m2 K m3 and K m4 are the Michaelis Menten constants defined as K m1 = k 1 + k 2 k 1 K m2 = k 3 + k 4 k 3 K m3 = h 1 + h 2 h 1 K m4 = h 3 + h 4 h 3. Now we need to find a proper set U R 2 satisfying assumptions A1 A4. Suppose that U has the form U = { (x 1 x 2 ) x 1 > σ x 2 > σ x 1 + x 2 < 1 + σ } for some positive σ and V is any bounded open set such that D ε is contained in U V then A1 follows naturally. Moreover if { } K m1 K m2 σ σ 0 := min S tot (K m1 + K m2 ) K m3 K m4 S tot (K m3 + K m4 ) A2 also holds. To check A4 let us look at the matrix: ( ) B1 (x) 0 B(x) := D y g 0 (x m 0 (x) 0) = 0 B 2 (x) where the column vectors of B 1 (x) are ( ) B1 1 (x) = k1 S tot x 1 (k 1 + k 2 ) k 3 S tot (1 x 1 x 2 ) ( B1 2 (x) = k 1 S tot x 1 and the column vectors of B 2 (x) are k 3 S tot (1 x 1 x 2 ) (k 3 + k 4 ) ) ( ) B2 1 (x) = h3 S tot (1 x 1 x 2 ) (h 3 + h 4 ) h 1 S tot x 2 B 2 2 (x) = ( h3 S tot (1 x 1 x 2 ) h 1 S tot x 2 (h 1 + h 2 ) If both matrices B 1 and B 2 have negative traces and positive determinants then A4 holds. The trace of B 1 is ). k 1 S tot x 1 (k 1 + k 2 ) k 3 S tot (1 x 1 x 2 ) (k 3 + k 4 ).

20 546 J Nonlinear Sci (2008) 18: It is negative provided that The determinant of B 1 is σ k 1 + k 2 + k 3 + k 4. S tot (k 1 + k 3 ) k 1 (k 3 + k 4 )S tot x 1 + k 3 (k 1 + k 2 )S tot (1 x 1 x 2 ) + (k 1 + k 2 )(k 3 + k 4 ). It is positive if (k 1 + k 2 )(k 3 + k 4 ) σ S tot (k 1 (k 3 + k 4 ) + k 3 (k 1 + k 2 )). The condition for B 2 can be derived similarly. To summarize if we take { σ = min σ 0 k 1 + k 2 + k 3 + k 4 S tot (k 1 + k 3 ) h 1 + h 2 + h 3 + h 4 S tot (h 1 + h 3 ) (k 1 + k 2 )(k 3 + k 4 ) S tot (k 1 (k 3 + k 4 ) + k 3 (k 1 + k 2 )) (h 1 + h 2 )(h 3 + h 4 ) S tot (h 1 (h 3 + h 4 ) + h 3 (h 1 + h 2 )) then the assumptions A1 A2 and A4 will hold. Notice that dy/dt in (12) is linear in y when ε = 0 so g 1 (defined as in (3)) is linear in z and hence the equation for z can be written as dz dτ = B(x 0)z x 0 U where the matrix B(x 0 ) is Hurwitz for every x 0 U. Therefore A3 also holds. It remains to show that assumption A6 is satisfied. Let us look at the reduced system (ε = 0in(12)): dx 1 dt dx 2 dt = = k 2 x 1 K m1 S tot + K + m1(1 x 1 x 2 ) K m2 + x 1 h 2 cx 2 K m3 S tot + K + m3(1 x 1 x 2 ) K m4 + x 2 h 4 c K m3(1 x 1 x 2 ) K m4 } K m3 S tot + K := F m3(1 x 1 x 2 ) 1 (x 1 x 2 ) K m4 + x 2 k 4 K m1 (1 x 1 x 2 ) K m2 K m1 S tot + K := F m1(1 x 1 x 2 ) 2 (x 1 x 2 ). K m2 + x 1 It is easy to see that F 1 is strictly decreasing in x 2 and F 2 is strictly decreasing in x 1 on K 0 = { (x 1 x 2 ) x 1 0x 2 0x 1 + x 2 1 }. The reduced system (13) is a strictly competitive system. By Theorem 1.1 of Hirsch (1985) flows of (13) have positive derivatives with respect to the cone { (x1 x 2 ) x 1 0x 2 0 } (13) and thus assumption A6 is satisfied. Applying Theorem 1wehave:

21 J Nonlinear Sci (2008) 18: Theorem 2 There exist a positive ε <ε 0 such that for each ε (0ε ) the forward trajectory of (12) starting from almost every point in D ε converges to some equilibrium. It is worth pointing out that the conclusion we obtained from Theorem 2 is valid only for small enough ε; that is the concentration of the enzyme should be much smaller than the concentration of the substrate. Unfortunately this is not always true in biological systems especially when feedbacks are present. However if the sum of the Michaelis Menten constants and the total concentration of the substrate are much larger than the concentration of enzyme a different scaling x 1 = [S 0] A x 2 = [S 2] A ε = E tot A t = τε where A = S tot +K m1 +K m2 +K m3 +K m4 allows us to obtain the same convergence result. 5.2 Another Example The following example demonstrates the importance of the smallness of ε. Consider an m + 1-dimensional system under the following assumptions: dx dt = γ(y 1...y m ) β(x) ε dy i = d i y i α i (x) d i > 0 i= 1...m dt 1. There exists an integer r>1such that the derivatives of γβ and α i are of class Cb r for sufficiently large bounded sets. 2. The function β(x) is odd and it approaches infinity as x approaches infinity. 3. The function α i (x) (i = 1...m) is bounded by positive constant M i for all x R. 4. The number of roots to the equation is countable. γ(α 1 (x)... α m (x)) = β(x) We are going to show that on any large enough region and provided that ε is sufficiently small almost every trajectory converges to an equilibrium. To emphasize the need for small ε we also show that when ε>1 limit cycles may appear. Assumption 4 implies A7 and because of the form of (14) A3 and A4 follow naturally. A6 also holds as every one-dimensional system is strongly monotone. For A5 we take D ε = { (x y) x a y i b i i = 1...m } (14)

22 548 J Nonlinear Sci (2008) 18: where b i is an arbitrary positive number greater than M i d i and a can be any positive number such that β(a) > N b := max γ(y 1...y m ). y i b i Picking such b i and a assures x dx dt < 0 y dy i i < 0 dt i.e. the vector fields point transversely inside on the boundary of D ε.letu and V be some bounded open sets such that D ε U V and assumption 1 holds on U and V. Then A1 and A2 follow naturally. By our main theorem for sufficiently small ε the forward trajectory of (14) starting from almost every point in D ε converges to some equilibrium. On the other hand convergence does not hold for large ε. Let β(x) = x3 3 x α 1(x) = 2 tanh x m = 1 γ(y 1 ) = y 1 d 1 = 1. It is easy to verify that (0 0) is the only equilibrium and the Jacobian matrix at (0 0) is ( ) /ε 1/ε When ε>1 the trace of the above matrix is 1 1/ε > 0 its determinant is 1/ε > 0 so the (only) equilibrium in D ε is repelling. On the other hand the set D ε is chosen such that the vector fields point transversely inside on the boundary of D ε.bythe Poincaré Bendixson Theorem there exists a limit cycle in D ε. 6 Conclusions Singular perturbation techniques are routinely used in the analysis of biological systems. The geometric approach is a powerful tool for global analysis since it permits one to study the behavior for finite ε on a manifold in which the dynamics are close to the slow dynamics. Moreover and most relevant to us a suitable fibration structure allows the tracking of trajectories and hence the lifting to the full system of the exceptional set of nonconvergent trajectories if the slow system satisfies the conditions of Hirsch s Theorem. Using the geometric approach we were able to provide a global convergence theorem for singularly perturbed strongly monotone systems in a form that makes it applicable to the study of double futile cycles. Acknowledgements We have benefited greatly from correspondence with Christopher Jones and Kaspar Nipp about geometric singular perturbation theory. We also wish to thank Alexander van Oudenaarden for questions that triggered much of this research and David Angeli Thomas Gedeon and Hal Smith for helpful discussions. This work was supported in part by NSF Grant DMS

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